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Mirrors > Home > MPE Home > Th. List > Mathboxes > un0.1 | Structured version Visualization version GIF version |
Description: ⊤ is the constant true, a tautology (see df-tru 1657). Kleene's "empty conjunction" is logically equivalent to ⊤. In a virtual deduction we shall interpret ⊤ to be the empty wff or the empty collection of virtual hypotheses. ⊤ in a virtual deduction translated into conventional notation we shall interpret to be Kleene's empty conjunction. If 𝜃 is true given the empty collection of virtual hypotheses and another collection of virtual hypotheses, then it is true given only the other collection of virtual hypotheses. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
un0.1.1 | ⊢ ( ⊤ ▶ 𝜑 ) |
un0.1.2 | ⊢ ( 𝜓 ▶ 𝜒 ) |
un0.1.3 | ⊢ ( ( ⊤ , 𝜓 ) ▶ 𝜃 ) |
Ref | Expression |
---|---|
un0.1 | ⊢ ( 𝜓 ▶ 𝜃 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | un0.1.1 | . . . 4 ⊢ ( ⊤ ▶ 𝜑 ) | |
2 | 1 | in1 39557 | . . 3 ⊢ (⊤ → 𝜑) |
3 | un0.1.2 | . . . 4 ⊢ ( 𝜓 ▶ 𝜒 ) | |
4 | 3 | in1 39557 | . . 3 ⊢ (𝜓 → 𝜒) |
5 | un0.1.3 | . . . 4 ⊢ ( ( ⊤ , 𝜓 ) ▶ 𝜃 ) | |
6 | 5 | dfvd2ani 39569 | . . 3 ⊢ ((⊤ ∧ 𝜓) → 𝜃) |
7 | 2, 4, 6 | uun0.1 39774 | . 2 ⊢ (𝜓 → 𝜃) |
8 | 7 | dfvd1ir 39559 | 1 ⊢ ( 𝜓 ▶ 𝜃 ) |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1654 ( wvd1 39555 ( wvhc2 39566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 199 df-an 386 df-tru 1657 df-vd1 39556 df-vhc2 39567 |
This theorem is referenced by: sspwimpVD 39915 |
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